Higher spin AdS$_{3}$ gravity and Tits-Satake diagrams

Abstract

We investigate higher spin AdS$_{3}$ gravity with real split forms of complex A$_{N}$ B$_{N}$, C$_{N}$ and D$_{N}$ Lie algebras. This is done by linking $SO(1,2)$ spin multiplets with splitted root systems using Tits-Satake diagrams of real forms. Unlike $SL(N,R)$, we show that the orthogonal families have two different higher spin (HS) spectrums: vectorial and spinorial. We find amongst others that the spinorial spectrum has an isolated spin j$_{\mathcal{N}}$ given by $\mathcal{N}\left(\mathcal{N}+1\right)/2$ for $SO(\mathcal{N},1+\mathcal{N})$ and $\mathcal{N} \left( \mathcal{N}-1\right) /2$ for $SO(\mathcal{N},\mathcal{N})$. We implement these results into the computation of the HS partition functions in these gravity theories and identify the individual contributions of the higher spin fields; valuable to manoeuver the HS-BTZ black hole partition function